Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is real-valued if and only if ...

A homework question asks me whether or not the matrix $W_N$ from the matrix representation of the Direct Fourier Transform is a Hermitian operator. From what I understand an Hermitian operator does ...

I wonder how I can write a function $f(\textbf{r})$ as a fourier series, when $f$ is periodic, in the sense that there exists a $ \textbf{T}_i \neq \textbf{0} $ so that $f(\textbf{r} + \textbf{T}_i) = ...

Fast Fourier transform (FFT) algorithms are able to calculate the discrete Fourier transform (DFT) in only $O(N\log N)$ asymptotical time. Since there is roughly $N\log N$ operations for computing $N$ ...

I am studying Fourier series right now. I asked a question before of math.statckexchange regarding Fourier series. This question is related and hopefully quite simple:
Generally Fourier series works ...

I am studying the Fourier series right now. Hopefully it's going okay. Now I have been playing a little bit with taking the product of a wave function (a sine or cosine with some phase) with a sine ...

Let $\ell^2(\mathbb{Z})$ be the set of all two-sided sequences $(a_i)$ in $\mathbb{C}$, such that $\sum_{n\in \mathbb{Z}} |a_n|^2 \lt \infty$.
What considerations do I have to take into account when ...

I sort of understand the principle of the Fourier series, but when I watch the wiki page I don't understand how to get from:
${a_0 \over 2} + \sum_{n=1}^N[a_n cos({2\pi n x \over P}) + b_n sin({2\pi ...